On operations preserving semi-transitive orientability of graphs
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by
Ilkyoo Choi, Jinha Kim, Minki Kim
2018
Abstract
We consider the class of semi-transitively orientable graphs, which is a much
larger class of graphs compared to transitively orientable graphs, in other
words, comparability graphs. Ever since the concept of a semi-transitive
orientation was defined as a crucial ingredient of the characterization of
alternation graphs, also knownas word-representable graphs, it has sparked
independent interest.
In this paper, we investigate graph operations and graph products that
preserve semitransitive orientability of graphs. The main theme of this paper
is to determine which graph operations satisfy the following statement: if a
graph operation is possible on a semitransitively orientable graph, then the
same graph operation can be executed on the graph while preserving the
semi-transitive orientability. We were able to prove that this statement is
true for edge-deletions, edge-additions, and edge-liftings. Moreover, for all
three graph operations,we showthat the initial semi-transitive orientation can
be extended to the new graph obtained by the graph operation.
Also, Kitaev and Lozin explicitly asked if certain graph products preserve
the semitransitive orientability. We answer their question in the negative for
the tensor product, lexicographic product, and strong product.We also push the
investigation further and initiate the study of sufficient conditions that
guarantee a certain graph operation to preserve the semi-transitive
orientability.
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